Journal
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS
Volume 148, Issue -, Pages -Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.jmps.2020.104264
Keywords
Localization; Elastic material; Finite strain; Asymptotic analysis; Energy methods
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The proposed method provides a way to derive equivalent one-dimensional models for slender non-linear structures, handling various conditions and capturing detailed shape of cross-sections through a kinematic parameterization. It is effective when macroscopic strain and rod properties vary slowly in the longitudinal direction, resulting in a more accurate equivalent rod model.
We propose a method for deriving equivalent one-dimensional models for slender non-linear structures. The approach is designed to be broadly applicable, and can handle in principle finite strains, finite rotations, arbitrary cross-sections shapes, inhomogeneous elastic properties across the cross-section, arbitrary elastic constitutive laws (possibly with low symmetry) and arbitrary distributions of pre-strain, including finite pre-strain. It is based on a kinematic parameterization of the actual configuration that makes use of a center-line, a frame of directors, and local degrees of freedom capturing the detailed shape of cross-sections. A relaxation method is applied that holds the framed center-line fixed while relaxing the local degrees of freedom; it is asymptotically valid when the macroscopic strain and the properties of the rod vary slowly in the longitudinal direction. The outcome is a one-dimensional strain energy depending on the apparent stretching, bending and twisting strain of the framed center-line; the dependence on the strain gradients is also captured, yielding an equivalent rod model that is asymptotically exact to higher order. The method is presented in a fully non-linear setting and it is verified against linear and weakly non-linear solutions available from the literature.
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